The present invention relates generally to data compression and expansion, and, more particularly, to the application of Gabor and logarithmic transforms to data compression and expansion.
Modern communications have a need to transmit increasingly large amounts of data through transmission channels that are constrained in time and frequency range. The general term for this problem is data compression and expansion, or xe2x80x9ccompanding.xe2x80x9d Algorithms and hardware that deal with companding must address the properties of images, audio, data, and RF communication signals. The present invention deals with a xe2x80x9clossyxe2x80x9d companding algorithm, which is suited to the transmission of images, audio, and RF signals, but not to binary computer data, where a lossy scheme can result in an unacceptable high bit error rate.
Sensor derived data often divides into two classes: those that provide constant absolute precision regardless of input level, and those where the desired data is actually a modulation of an underlying brightness, generating a constant relative precision. Most images are high frequency contrast modulations of an underlying scene brightness, where the contrast modulation remains much smaller than the total brightness range seen over the entire observation time. Similarly, most RF signals are intentionally modulated as the mathematical product of the information and an underlying carrier, whose brightness can vary dramatically upon reception. Therefore, the great majority of practical sensor applications are multiplicative modulation processes, and only require constant relative precision. This, in turn, implies that either floating point or logarithmic representations may be used without loss of resolution, as long as they are properly scaled.
The concept of applying such a logarithmic transform to a complex spectrum is well known, and is known as a homomorphic process. But most of the reported work is directed to the separation of multiplicatively combined signals, or echo removal, and not to compression of a signal. The basic homomorphic process consists of applying a logarithmic transform to the output of a Fourier transform in the forward direction, and inverting this sequence in the reverse direction, after suitable manipulations in the log Gabor domain. The purposes of the Fourier transform are to convert convolution processes to multiplications in frequency space, and to separate various signals that overlap in the time series, but which are separate in the frequency space. The log transform then allows the multiplicative signals still contained in each frequency sample to be treated as additive superposition. This process has not been suggested for use in companding operations.
Digital voice grade channels are commonly companded in commercial telephony to compress 12 bits of inherent dynamic range to an 8 bit format. These schemes are driven by a need for low complexity and cost, and take advantage of the characteristics of audio reception by a human ear. The formal systems that implement audio log compression are known as the Mu Law under a system developed by Bell Telephone or the A Law under CCIR in Europe. Each of these systems has the same basic characteristics:
a. Linear behavior for signal levels near zero.
b. Log behavior above a corner value, to a maximum input value set by the input digitizer.
An important liability of these pure logarithmic companding schemes is that they are performed on the raw time series, generating intermodulation products between all frequencies in the band. Some of these products fall out of band in the compressed channel, and are lost or are aliased down to new frequencies. In other words, the homomorphic process is not used and the absence of a Fourier Transform leaves multiple superposed signals to be log transformed together, resulting in destruction of the smaller signals in favor of the large signals. Even though this limitation is present, logarithmic companding is the rule for single digital voice grade channels, and relies on the fact that these channels are not designed to carry more than one speaker on each end of a telephone link.
One prior scheme (Lundquist et al., A Homomorphic Approach to Companding, 28th Asilomar Conf. on Signals, Systems and Computers (Nov. 1994)) combines companding and homomorphic processing to provide a compression/expansion process with homomorphic processes nested inside the companding. That is, a signal is compressed using an Mu/A law compressor and then subjected to Fourier and log transforms and then expanded by performing an inverse Fourier and log transform followed by a Mu/A law expansion.
But this process still provides for log compression before any Fourier transform has occurred so that the signal components will have the same mixing and suppression problems as the standard Mu/A Law systems. Further, the process now involves logarithms of logarithmic data, which is a very lossy process compared to a single logarithmic transformation. According to Lundquist, the addition of a complex homomorphic process does little to enhance the basic Mu/A Law performance.
In accordance with the present invention, a new homomorphic process is provided that enhances the basic Mu/A Law performance of a logarithmic companding system.
Various objects, advantages and novel features of the invention will be set forth in part in the description which follows, and in part will become apparent to those skilled in the art upon examination of the following or may be learned by practice of the invention. The objects and advantages of the invention may be realized and attained by means of the instrumentalities and combinations particularly pointed out in the appended claims.
In accordance with the purposes of the present invention, as embodied and broadly described herein, the present invention includes a method for logarithmic compression of time series data. A log Gabor transformation is made of incoming time series data to output phase and logarithmic magnitude spectrum values. The output phase and logarithmic magnitude spectrum values are compressed by selecting only magnitude values above a selected threshold and corresponding phase values to transmit as compressed phase and logarithmic magnitude spectrum values. The transmitted values are expanded and a reverse log Gabor transformation is then performed on the transmitted phase and logarithmic magnitude spectrum values to output transmitted time series data to a user.
In a particular aspect of the present invention, the log Gabor transformation is performed by forming a Fourier transform of input time data to output spectral data as frequency domain data, and then transforming the spectral data in frequency domain format to polar coordinate format having magnitude and phase information. The magnitude information is finally transformed to a logarithmic format to output an output logarithmic magnitude value and a phase value.